Computing and Visualization in Science

, Volume 15, Issue 2, pp 75–86 | Cite as

Tensor structured evaluation of singular volume integrals

  • Jonas Ballani
  • Peter Meszmer


In this article, we introduce a new method for the accurate and fast computation of singular integrals over cuboids in three-dimensional space. Using a straightforward geometric parametrisation of the domain of integration, we interpret the integral as a smooth function on a high-dimensional parameter space. A standard interpolation scheme then leads to a high-dimensional tensor to which an approximation in the data-sparse hierarchical tensor format is applied. Once this approximation is available, the evaluation of an integral value becomes an easy task which does no longer require the treatment of singular terms. Numerical experiments illustrate the potential of the proposed approach for typical examples.


  1. 1.
    Ballani, J.: Fast evaluation of singular BEM integrals based on tensor approximations. Numer. Math. 121(3), 433–460 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical Tucker format. Linear Algebra Appl. 438(2), 639–657 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertoglio, C., Khoromskij, B.N.: Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels. Comput. Phys. Comm. 183(4), 904–912 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Börm, S., Hackbusch, W.: Hierarchical quadrature of singular integrals. Computing 74, 75–100 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hackbusch, W.: Direct integration of the Newton potential over cubes. Computing 68(3), 193–216 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hackbusch, W.: Entwicklungen nach Exponentialsummen. Preprint 4/2005. Max Planck Institute for Mathematics in the Sciences (2005)Google Scholar
  8. 8.
    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals. J. Numer. Math. 18(2), 91–117 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals—part II. Preprint 71/2012. Max Planck Institute for Mathematics in the Sciences (2012)Google Scholar
  12. 12.
    Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31(5), 3744–3759 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Tobler, C.: Low-rank tensor methods for linear systems and eigenvalue problems. Ph.D. thesis, ETH Zürich (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations